Abstract

In the framework of the scale relativity theory, the chaotic behavior in time only of a number of macroscopic systems corresponds to the motion in a space with geodesics of fractal dimension 2 and leads to its representation by a Schrödinger-type equation acting in the macroscopic domain. The fluid interpretation of such a Schrödinger equation yields Euler and Navier–Stokes equations. We therefore choose to extend this formalism to study the properties of a system exhibiting a chaotic behavior both in space and time, which amounts to consider them as issued from the geodesic features of a mathematical object exhibiting all the properties of a fractal “space-time.” Starting with the simplest Klein–Gordon-type form that can be given to the geodesic equation in this case, we obtain a motionequation for a “three fluid” velocity field and three continuity equations, together with parametric expressions for the three velocity components which allow us to derive relations between their nonvanishing curls. At the nonrelativistic limit and owing to the physical properties exhibited by this solution, we suggest that it could represent some kind of three-dimensional chaotic behavior in a classical fluid, tentatively turbulent if particular conditions are fulfilled. The appearance of a transition parameter in the equations allows us to consider different ways of testing experimentally our proposal.

Received 10 July 2008Accepted 10 November 2009Published online 18 December 2009

Acknowledgments:

The author wants to thank Thierry Lehner for valuable discussions and comments about this work and Laurent Nottale for the communication of material and results unpublished at the time this article was submitted for publication.